An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning

Abstract

This paper extends an economic order quantity (EOQ) model for items with imperfect quality based on two different holding costs and learning considerations. This is one of the few attempts aiming at combining the EOQ model, learning theory, and fuzzy technique in solving an EOQ problem. In present research, a fuzzy model is developed in which both parameters and decision variables are fuzzified and represented by triangular fuzzy numbers (TFNs). The total profit per unit time is obtained using fuzzy arithmetic operations, and then defuzzified by the graded mean integration value (GMIV) method. Using Karush-Kuhn-Tucker (KKT) conditions, the optimal lot size is obtained from the defuzzified total profit per unit time function. A numerical example for investigating the behavior of the model in a fuzzy situation is presented, and directions for future study are proposed. Besides, the results of the developed fully fuzzy model are compared with some previous ones in the literature.

Keywords

EOQ model imperfect quality holding cost learning effect triangular fuzzy number graded mean integration value method

1 Introduction

In today’s competitive market, inventory management is certainly one of the most challenging problems for organizations. This is a sequel to the fact that keeping inventory is tied up in the financial resources of organizations. The importance of inventory management has been growing recently as it has been found that the agility of organizations highly depends on the efficient management of inventories. In this regard, organizations try to manage their stock efficiently which requires keeping safety stocks for situations where unexpected events such as fluctuations in demand, market structural changes and machinery deficiencies are experienced.

The need for developing a mathematical model to cope with inventory management problems dates back to the early twentieth century when Harris [18] and Taft [47] developed the first inventory models. Despite simple assumptions and uncomplicated mathematics, the models were widely accepted in many practical situations. However, the lack of effective assumptions, which leds to the inability of the models in real applications, is the main weaknesses. For instance, there is the likelihood that some defective products are produced and/or received in an inventory system, while the basic models assume that all produced or received lots are of perfect quality.

A problem that frequently appears in inventory decision-making is impreciseness in defining some or all components of a model. In order to cope with the impreciseness of input parameters in realistic environments, the Fuzzy Set Theory (FST), introduced by Zadeh [56], has been recognized as a powerful tool and received considerable attention from researchers [12 , 46]. The FST has been identified as a proper tool in handling decision-making under uncertainty, where the information available to the DM is incomplete. Therefore, the DM may use his experience and prefer to describe imprecise data linguistically. When it comes to inventory management, there are many situations in industry where the attributes of the input parameters are complicated so that they can be determined based on the experience of the DM. The application of the FST in inventory management can provide flexibility in defining the vague parameters, which enables the model to account for uncertainties.

Another shortcoming in the earlier inventory models is that they would not consider the learning aspect. In real production systems, as time goes by, the knowledge and experience of workers about operations and processes increase naturally and lead to improvements in their performance. For example, as a result of learning in a production system, the rate of manufacturing of defective items may reduce. The learning curve concept, originally presented by Wright [52], is defined as a natural phenomenon that occurs when a worker performs a task repetitively.

Among various studies published with respect to the application of FST and learning theory in inventory management, only rare cases could be found that simultaneously integrate both concepts in an inventory model. This paper aims to investigate how learning and fuzziness concepts influence the optimal policy of an EOQ model containing imperfect quality items. This model, to the best of our knowledge, is the first EOQ model developed with learning influence on the percentage of defective items with entirely imprecise parameters. To model impreciseness, the inventory parameters are defined with a degree of uncertainty. In order to be more compatible with real imperfect supply case, this study also differentiates between the holding cost of good and imperfect items.

The next section offers a brief review of the previous works. Section 3 provides a brief description about perquisite fuzzy mathematics. Section 4 reviews the Wahab and Jaber’s model [51] that is developed in a fully fuzzy environment in Section 5. Numerical examples are presented in Section 6. Comparison with previous models and managerial insights are discussed in Section 7 and 8, respectively. Finally, the paper is concluded in Section 9.

2 Literature review

Application of the FST in inventory management has become an interesting topic of research in recent years, and there are numerous works in the literature that have investigated fuzzifying some parameters or variables of various classical models.

Lee and Yao [37] investigated a fuzzy EOQ model with two triangular fuzzy parameters: (1) demand, and (2) production quantities. Lin and Yao [38] extended the work of Lee and Yao [37] by fuzzifying the production quantity. Hsieh [20] studied the problem of determining the optimal production quantity in a fuzzy production inventory model. Chang [8] fuzzified the work of Salameh and Jaber [42] who assumed that a lot containing a fixed percentage of imperfect quality items is subjected to a 100% screening while non-conforming items are withdrawn from the inventory and sold as a single batch at a discounted price. Chang et al. [9] extended the fuzzy modification of the model proposed by Ouyang and Yao [39] by considering the lead time as a random variable assumed as a TFN. Yazgi Tütüncü et al. [55] incorporated the FST into a classical continuous inventory model with probabilistic demand where all costs of the model were assumed to be fuzzy numbers. Considering the fuzziness of the cost, Vijayan and Kumaran [49] proposed an inventory model with a mixture of backorders and lost sales in a fuzzy situation. Along with their previous works, Vijayan and Kumaran [50] modified the so-called classic economic order time model by examining it under various fuzzy conditions. Björk [5] addressed an inventory model with backorders through considering both demand and lead time as uncertain parameters. Kazemi et al. [31] reconsidered the model studied in Björk [5] and proposed an analytical solution for solving the fully fuzzy case of the model using a different approach, where the model was tested under both triangular and trapezoidal fuzzy numbers. Baykasoglu and Gocken [3] developed a constrained multi-item EOQ model where all parameters of the problem were modeled with the help of TFN. Björk [6] presented an inventory model with a finite production rate and without shortage. The reader is referred to Guiffrida [17] for a reasonably comprehensive review of fuzzy inventory models. Shekarian et al. [45] developed a fuzzified economic production quantity (EPQ) model that generates defective items in a single-stage production system with planned backorders, which is found to be a fuzzified version of C $\overset{´}{a}$ rdenas-Barrón [7]. Moreover, Shekarian et al. [44] developed a single-stage production system fuzzifying the rate of defectives and the demand rate as TFN.

Previous researches investigated the application of the learning theory in some aspects of the operations management including inventory management. Research in this field includes but is not limited to Balkhi [2], Jaber et al. [21, 22], Jaber and Bonney [24, 25], Khan et al. [34], Wahab and Jaber [51]. Readers are referred to Jaber [23] and Grosse et al. [16] for a broad discussion on the application of the learning theory in logistics, production, and inventory systems. In addition, Glock [13] investigated the effect of learning on the supplier selection problem. Glock and Jaber [14] developed a multi-stage production-inventory system considering learning and forgetting in production and rework processes which a proportion of defective items is produced at each stage. Furthermore, Jaber and Glock [26] developed a new learning curve model, which was investigated by incorporating it into an EPQmodel.

Although some papers in the literature have considered the concept of learning and FST simultaneously, the number of such studies is limited. Bera et al. [4] developed an EPQ model for a deteriorating item and assumed that learning affects the set up cost.Pal et al. [40] studied an EPQ model with fuzzy life time, time, and price dependent demand. Yadav et al. [54] developed an inventory model with fuzzy demand and learning in holding and ordering costs and number of defective items, and investigated the effect of fuzziness and learning on lot size and profit. Although their model contained imperfect quality items, it only fuzzi-fied demand and would not differentiate between the holding cost of imperfect and perfect items. In this line of research, the studies of Glock et al. [15], Kazemi et al. [32, 33] are considerably different from those described above. In these researches, the possibility of reducing the fuzziness in demand in an EOQ model using the learning curve was illustrated. For a comprehensive review on the application of learning curves in various research areas, readers are referred to Anzanello and Fogliatto [1]. In addition, Kumar and Goswami [35], Pathak et al. [41], Yadav et al. [53] presented the most recent work in this context. Table 1 compares some of the main characteristics of the investigated researches to highlight the differences among these studies and the present paper.

3 Fuzzy preliminaries

DMs have to estimate inventory parameters commonly in the cases that they are dealing with ambiguity or lack of perfect information. It has been proven in literature that the application of the FST significantly improves the precision in setting the model parameters. The FST has noticeable flexibility in real situations as it allows the model to easily incorporate various subjective opinions of experts in parameter’s estimation [57].

Definition 2. A fuzzy subset $\tilde{A}$ of the universe set X is normal if and only if $sup_{x \in X} μ_{\tilde{A}} (x) = 1$ . That is, the largest grade that an element can obtain.

Definition 3. A fuzzy subset $\tilde{A}$ of the universe set X is convex if and only if:

$μ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) \geq min (μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2}))$ for all x₁, x₂ ∈ X and for λ ∈ [0, 1].

Definition 4. A fuzzy set $\tilde{A}$ is a fuzzy number if and only if $\tilde{A}$ is normal and convex on X.

The TFN is identified as the most popular fuzzy number. It is specified by the triplet (a₁, a₂, a₃) where a₁ ≤ a₂ ≤ a₃ and a₁, a₂, a₃ ∈ R.

3.1 Function principle

Function Principle, originally developed by Chen [10], is one of the commonest type of fuzzy arithmetical operation in application of the FST in inventory management. This method does not change the shape of linear membership functions. It is an appropriate fuzzification method for the model of interest, since the developed method involves a couple of fuzzy multiplication terms, which makes the model more complex. Therefore, it is utilized to make the handling of fuzzy arithmetic operations feasible as well as to prevent obtaining a degenerated solution.

Let $\tilde{A} = (a_{1}, a_{2}, a_{3})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3})$ be two positive TFNs. Then, based on the Functional Principle, the operations of the fuzzy numbers $\tilde{A}$ and $\tilde{B}$ are expressed as: $\begin{matrix} \tilde{A} + \tilde{B} & = & (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}), \\ \tilde{A} - \tilde{B} & = & (a_{1} - b_{3}, a_{2} - b_{2}, a_{3} - b_{1}), \\ \tilde{A} \times \tilde{B} & = & (a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3}), \\ \frac{\tilde{A}}{\tilde{B}} & = & (\frac{a_{1}}{b_{3}}, \frac{a_{2}}{b_{2}}, \frac{a_{3}}{b_{1}}) . \end{matrix}$

3.2 Graded Mean Integration Value (GMIV) of fuzzy numbers

Chen [10] and Chen and Hsieh [11] introduced the GMIV method based on the integral value of graded mean α-level of fuzzy numbers. The GMIV of TFN $\tilde{A}$ is denoted by $Φ (\tilde{A})$ and defined as:

$\begin{matrix} Φ (\tilde{A}) & = & \int_{0}^{1} \frac{α (L^{- 1} (α) + R^{- 1} (α))}{2} d α / \int_{0}^{1} α d α \\ = & \int_{0}^{1} α (L^{- 1} (α) + R^{- 1} (α)) d α, \end{matrix}$ (1) where L^-1, R^-1 are respectively inverse functions of L and R, and the graded α-level value of $\tilde{A}$ is: $[α (L^{- 1} (α) + R^{- 1} (α))] / 2$

4 Classical EOQ model for items with imperfect quality, different holding costs, and learning effects

Wahab and Jaber [51] developed an EOQ model for an inventory system based on the models presented by Salameh and Jaber [42] and Jaber et al. [22], where the shipped lot contain defective items under learning consideration. They assumed that the holding costs for good and defective items are not the same. They considered an inventory system in which the shipped lot contains defective items with a known probability distribution. The model includes the following notations and assumptions:

Assumptions

Demand rate is constant during the planninghorizon

Shortages are not allowed

Lead time is zero

Each shipment undergoes 100% inspectionprocess

Defective items are sold at a discounted price

Percentage of defectives items follows a learning curve

Time horizon is infinite/finite

Notations

Demand per year

Lot size

Unit variable cost

Fixed cost of placing an order

Unit selling price of items of good quality

Unit selling price of defective items, v < s

Screening rate

Unit screening cost

Screening time

Percentage of defective items per shipment

Probability density function of p

Cycle length

Holding cost for good items per unit of time

Holding cost for defective items per unit of time

If we assume that p (n), which is defined as percentage of defective items per shipment, follows the below S-shaped logistic learning curve model: $p (n) = \frac{α}{γ + e^{β n}}$ where α, β and γ are the parameters of the learning function. Then, according to Wahab and Jaber [51], the total profit per unit time TPU and the optimal lot size $y_{n}^{*}$ , are as below: $\begin{matrix} TPU (y) = D (s - v + \frac{h_{g} y}{2 x} + \frac{h_{d} y}{2 x}) \\ + (\frac{D}{1 - p (n)}) (v - \frac{K}{y} - c - d - \frac{h_{g} y}{2 x} - \frac{h_{d} y}{2 x}) \\ - \frac{h_{g} y (1 - p (n))}{2} \end{matrix}$ (2) $y_{n}^{*} = \sqrt{\frac{Ξ_{1}}{Ξ_{2}}}$ (3) where $\begin{matrix} Ξ_{1} & = & 2 DK [1 / (1 - p (n))] \\ Ξ_{2} & = & h_{g} (1 - p (n)) \\ + (D / x) (h_{g} + h_{d}) [1 / (1 - p (n))] \\ - (D / x) (h_{g} + h_{d}) \end{matrix}$

5 Integrating the concept of FST into the considered EOQ model

In this section, the crisp model presented in Section 4 is fully fuzzified by fuzzifying all the input parameters (D, s, v, h_g, h_d, p (n) , x, K, c, d) and variable (y). It is assumed that all the input parameters are uncertain. The uncertainty of the considered parameters is represented by the concept of the TFN, as follows: $\begin{matrix} \tilde{D} & = & (D - Δ_{l}^{D}, D, D + Δ_{h}^{D}), \\ \tilde{s} & = & (s - Δ_{l}^{s}, s, s + Δ_{h}^{s}), \end{matrix}$

$\begin{matrix} \tilde{ν} & = & (v - Δ_{l}^{v}, v, v + Δ_{h}^{v}), \\ {\tilde{h}}_{g} & = & (h_{g} - Δ_{l}^{h_{g}}, h_{g}, h_{g} + Δ_{h}^{h_{g}}), \\ {\tilde{h}}_{d} & = & (h_{d} - Δ_{l}^{h_{d}}, h_{d}, h_{d} + Δ_{h}^{h_{d}}), \\ \tilde{p} (n) & = & (p (n) - Δ_{l}^{p (n)}, p (n), p (n) + Δ_{h}^{p (n)}), \\ \tilde{x} & = & (x - Δ_{l}^{x}, x, x + Δ_{h}^{x}), \\ \tilde{K} & = & (k - Δ_{l}^{k}, k, k + Δ_{h}^{k}), \\ \tilde{c} & = & (c - Δ_{l}^{c}, c, c + Δ_{h}^{c}), \\ \tilde{d} & = & (d - Δ_{l}^{d}, d, d + Δ_{h}^{d}) . \end{matrix}$

Moreover, the decision variable is assumed to be a TFN and is given by: ${\tilde{y}}_{n} = (y_{n} - Δ_{l}^{y_{n}}, y_{n}, y_{n} + Δ_{h}^{y_{n}}),$ Note that: $\begin{matrix} Δ_{l}^{i} > 0; i & = & D, s, v, h_{g}, h_{d}, p (n), x, K, c, d, y_{n}; \\ Δ_{h}^{j} > 0; j & = & D, s, v, h_{g}, h_{d}, p (n), x, K, c, d, y_{n} \end{matrix}$ and: $\begin{matrix} D > Δ_{l}^{D}, s > Δ_{l}^{s}, v > Δ_{l}^{v}, h_{g} > Δ_{l}^{h_{g}}, h_{d} > Δ_{l}^{h_{d}}, \\ p (n) > Δ_{l}^{p (n)}, x > Δ_{l}^{x}, K > Δ_{l}^{K}, c > Δ_{l}^{c}, d > Δ_{l}^{d}, \\ y_{n} > Δ_{l}^{y_{n}} \end{matrix}$

In addition, $Δ_{l}^{i}$ and $Δ_{h}^{j}$ are arbitrary values that can be determined through expert knowledge and the historical background of the inventory parameters. For example, the value of $D - Δ_{l}^{D}$ can be considered as the smallest value that is observed for the demand in previous inventory planning horizons, and, therefore, $Δ_{l}^{D}$ will be the deviation of the smallest value of the demand from the most promising value. By replacing the above TFN in the TPU formula, the fuzzy total profit per unit time can be obtained as follows:

$\begin{matrix} T \tilde{PU} ({\tilde{y}}_{n}) = \tilde{D} (\tilde{s} - \tilde{ν} + \frac{{\tilde{h}}_{g} {\tilde{y}}_{n}}{2 \tilde{x}} + \frac{{\tilde{h}}_{d} {\tilde{y}}_{n}}{2 \tilde{x}}) \\ + (\frac{\tilde{D}}{1 - \tilde{p} (\tilde{n})}) (\tilde{ν} - \frac{\tilde{K}}{\tilde{y}} - \tilde{c} - \tilde{d} - \frac{{\tilde{h}}_{g} {\tilde{y}}_{n}}{2 \tilde{x}} - \frac{{\tilde{h}}_{d} {\tilde{y}}_{n}}{2 \tilde{x}}) \\ - \frac{{\tilde{h}}_{g} {\tilde{y}}_{n} (1 - \tilde{p} (\tilde{n}))}{2} \end{matrix}$ (4) we define the following symbols: $ξ = \tilde{D} (\tilde{s} - \tilde{ν} + \frac{{\tilde{h}}_{g} {\tilde{y}}_{n}}{2 \tilde{x}} + \frac{{\tilde{h}}_{d} {\tilde{y}}_{n}}{2 \tilde{x}})$ (5) $ζ = (\tilde{ν} - \frac{\tilde{K}}{{\tilde{y}}_{n}} - \tilde{c} - \tilde{d} - \frac{{\tilde{h}}_{g} {\tilde{y}}_{n}}{2 \tilde{x}} - \frac{{\tilde{h}}_{d} {\tilde{y}}_{n}}{2 \tilde{x}})$ (6) $ς = (\frac{\tilde{D}}{1 - \tilde{p} (\tilde{n})})$ (7) $τ = [{\tilde{h}}_{g} {\tilde{y}}_{n} (1 - \tilde{p} (\tilde{n}))] / 2$ (8)

Since all the parameters are TFNs, they can be considered as a TFN by the following triplet components according to the fuzzy arithmetic operations described in Section 3.1:

$\begin{matrix} ξ_{1} = (D - Δ_{l}^{D}) [[(s - Δ_{l}^{s}) - (v + Δ_{h}^{v})] \\ + \frac{(h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}}) + (h_{d} - Δ_{l}^{h_{d}}) (y_{n} - Δ_{l}^{y_{n}})}{2 (x + Δ_{h}^{x})}] \end{matrix}$ (9) $ξ_{2} = D [(s - v) + \frac{h_{g} y_{n}}{2 x} + \frac{h_{d} y_{n}}{2 x}]$ (10) $\begin{matrix} ξ_{3} = (D + Δ_{h}^{D}) [[(s + Δ_{h}^{s}) - (v - Δ_{l}^{v})] \\ + \frac{(h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}}) + (h_{d} + Δ_{h}^{h_{d}}) (y_{n} + Δ_{h}^{y_{n}})}{2 (x - Δ_{l}^{x})}] \end{matrix}$ (11) $\begin{matrix} ζ_{1} = (v - Δ_{l}^{v}) - \frac{(K + Δ_{h}^{K})}{(y_{n} - Δ_{l}^{y_{n}})} - (c + Δ_{h}^{c}) - (d + Δ_{h}^{d}) \\ - \frac{((h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}}) + (h_{d} + Δ_{h}^{h_{d}}) (y_{n} + Δ_{h}^{y_{n}})}{2 (x - Δ_{l}^{x})} \end{matrix}$ (12) $ζ_{2} = (v - \frac{K}{y_{n}} - c - d - \frac{h_{g} y_{n}}{2 x} - \frac{h_{d} y_{n}}{2 x})$ (13) $\begin{matrix} ζ_{3} = (v + Δ_{h}^{v}) - \frac{(K - Δ_{l}^{K})}{(y_{n} + Δ_{h}^{y_{n}})} - (c - Δ_{l}^{c}) - (d - Δ_{l}^{d}) \\ - \frac{((h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}}) + (h_{d} - Δ_{l}^{h_{d}}) (y_{n} - Δ_{l}^{y_{n}})}{2 (x + Δ_{h}^{x})} \end{matrix}$ (14)

$\begin{matrix} ς_{1} = (D - Δ_{l}^{D}) \\ [\frac{(γ + Δ_{h}^{γ}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}{(γ + Δ_{h}^{γ}) - (α - Δ_{l}^{α}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}] \end{matrix}$ (15) $ς_{2} = (\frac{γ + e^{n β}}{γ + e^{n β} - α}) D$ (16)

$\begin{matrix} ς_{3} = (D + Δ_{h}^{D}) \\ [\frac{(γ - Δ_{l}^{γ}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}{(γ - Δ_{l}^{γ}) - (α + Δ_{h}^{α}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}] \end{matrix}$ (17)

$\begin{matrix} τ_{1} = [\frac{(h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}})}{2}] \\ [\frac{(γ - Δ_{l}^{γ}) - (α + Δ_{h}^{α}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}{(γ - Δ_{l}^{γ}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}] \end{matrix}$ (18) $τ_{2} = (\frac{h_{g} y_{n}}{2}) (\frac{γ - α + e^{n β}}{γ + e^{n β}})$ (19)

$\begin{matrix} τ_{3} = [\frac{(h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}})}{2}] \\ [\frac{(γ + Δ_{h}^{γ}) - (α - Δ_{l}^{α}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}{(γ + Δ_{h}^{γ}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}] \end{matrix}$ (20)

Again, because all of the parameters of the inventory system are TFNs, the TPU could be a TFN expressed as below: $\begin{matrix} T \tilde{PU} ({\tilde{y}}_{n}) = (C_{1}, C_{2}, C_{3}) = \\ ((ξ_{1} + ς_{1} ζ_{1} - τ_{3}), (ξ_{2} + ς_{2} ζ_{2} - τ_{2}), (ξ_{3} + ς_{3} ζ_{3} - τ_{1})) \end{matrix}$ (21)

The GMIV of TPU could be expressed as follows:

$\begin{matrix} Φ (T \tilde{PU} ({\tilde{y}}_{n})) & = & \frac{1}{6} (ξ_{1} + ς_{1} ζ_{1} - τ_{3}) \\ + \frac{4}{6} (ξ_{2} + ς_{2} ζ_{2} - τ_{2}) \\ + \frac{1}{6} (ξ_{3} + ς_{3} ζ_{3} - τ_{1}) \end{matrix}$ (22)

By replacing the obtained terms, we have:

$\begin{matrix} Φ (T \tilde{PU} ({\tilde{y}}_{n})) & = & \frac{1}{6} ((D - Δ_{l}^{D}) [[(s - Δ_{l}^{s}) - (v + Δ_{h}^{v})] \\ + \frac{(h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}}) + (h_{d} - Δ_{l}^{h_{d}}) (y_{n} - Δ_{l}^{y_{n}})}{2 (x + Δ_{h}^{x})}] \\ + (D - Δ_{l}^{D}) [\frac{(γ + Δ_{h}^{γ}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}{(γ + Δ_{h}^{γ}) - (α - Δ_{l}^{α}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}] \\ [\begin{matrix} (v - Δ_{l}^{v}) - \frac{(K + Δ_{h}^{K})}{(y_{n} - Δ_{l}^{y_{n}})} - (c + Δ_{h}^{c}) - (d + Δ_{h}^{d}) \\ - \frac{((h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}}) + (h_{d} + Δ_{h}^{h_{d}}) (y_{n} + Δ_{h}^{y_{n}})}{2 (x - Δ_{l}^{x})} \end{matrix}] \\ - [\frac{(h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}})}{2}] [\frac{(γ + Δ_{h}^{γ}) - (α - Δ_{l}^{α}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}{(γ + Δ_{h}^{γ}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}]) \\ + \frac{4}{6} (D [(s - v) + \frac{h_{g} y_{n}}{2 x} + \frac{h_{d} y_{n}}{2 x}] \\ + D (\frac{γ + e^{n β}}{γ + e^{n β} - α}) (v - \frac{K}{y_{n}} - c - d - \frac{h_{g} y_{n}}{2 x} + \frac{h_{d} y_{n}}{2 x}) \\ - (\frac{h_{g} y_{n}}{2}) (\frac{γ - α + e^{n β}}{γ + e^{n β}})) + \frac{1}{6} ((D + Δ_{h}^{D}) [[(s + Δ_{h}^{s}) - (v - Δ_{l}^{v})] \\ + \frac{(h_{g} + Δ_{h}^{h_{g}}) (y_{n} + Δ_{h}^{y_{n}}) + (h_{d} + Δ_{h}^{h_{d}}) (y_{n} + Δ_{h}^{y_{n}})}{2 (x - Δ_{l}^{x})}] \\ + (D + Δ_{h}^{D}) [\frac{(γ - Δ_{l}^{γ}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}{(γ - Δ_{l}^{γ}) - (α + Δ_{h}^{α}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}] \\ [\begin{matrix} (v + Δ_{h}^{v}) - \frac{(K - Δ_{l}^{K})}{(y_{n} + Δ_{h}^{y_{n}})} - (c - Δ_{l}^{c}) - (d - Δ_{l}^{d}) \\ - \frac{((h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}}) + (h_{d} - Δ_{l}^{h_{d}}) (y_{n} - Δ_{l}^{y_{n}})}{2 (x + Δ_{h}^{x})} \end{matrix}] \\ - [\frac{(h_{g} - Δ_{l}^{h_{g}}) (y_{n} - Δ_{l}^{y_{n}})}{2}] \\ [\frac{(γ - Δ_{l}^{γ}) - (α + Δ_{h}^{α}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}{(γ - Δ_{l}^{γ}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}]) \end{matrix}$ (23) Before initializing the process of finding the optimal solution for the defuzzified inventory cost function, consider the following equations: ${\begin{matrix} y_{n} - Δ_{l}^{y_{n}} = y_{1} \\ y_{n} = y_{2} \\ y_{n} + Δ_{h}^{y_{n}} = y_{3} \end{matrix}$ (24) $Φ (T \tilde{PU} ({\tilde{y}}_{n}))$ , is taken as the crisp estimate of $T \tilde{PU} ({\tilde{y}}_{n})$ where it is optimized subject to the following condition: 0 < y₁ ≤ y₂ ≤ y₃.

Thus, the optimum solution of $Φ (T \tilde{PU} ({\tilde{y}}_{n}))$ can be found by optimizing Equation (23) subject to the following inequality constraints: $y_{1} - y_{2} \leq 0, y_{2} - y_{3} \leq 0, - y_{1} < 0,$

The KKT theorem discussed by Taha [48] and Hillier and Lieberman [19] is then used to find the optimal solution of $Φ (T \tilde{PU} ({\tilde{y}}_{n}))$ subject to the following inequalities as imposed conditions. $y_{1} (\frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{1}} - λ_{1} + λ_{3}) = 0$ (25) $y_{2} (\frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{2}} + λ_{1} - λ_{2}) = 0$ (26) $y_{3} (\frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{3}} + λ_{2}) = 0$ (27) $y_{1} - y_{2} \leq 0$ (28) $y_{2} - y_{3} \leq 0$ (29) $- y_{1} < 0$ (30) $λ_{1} (y_{1} - y_{2}) = 0$ (31) $λ_{2} (y_{2} - y_{3}) = 0$ (32) $λ_{3} (- y_{1}) = 0$ (33) $y_{1} \geq 0, y_{2} \geq 0, y_{3} \geq 0$ (34) $λ_{1} \geq 0, λ_{2} \geq 0, λ_{3} \geq 0$ (35)

where the derivatives of $T \tilde{PU} ({\tilde{y}}_{n})$ with respect to y₁, y₂ and y₃ are as below:

$\begin{matrix} \frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{1}} & = & \frac{1}{6} [(D - Δ_{l}^{D}) (A + \frac{(K + Δ_{h}^{K})}{y_{1}^{2}} B) \\ - A (D + Δ_{h}^{D}) C - \frac{(h_{g} - Δ_{l}^{h_{g}})}{2 C}] \end{matrix}$ (36) $\frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{2}} = \frac{4}{6} [DF + (\frac{K}{y_{2}^{2}} - F) D G - \frac{h_{g}}{2 G}]$ (37)

$\begin{matrix} \frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{3}} & = & \frac{1}{6} [\frac{(h_{g} + Δ_{h}^{h_{g}})}{2 B} - (D - Δ_{l}^{D}) BE \\ + (D + Δ_{h}^{D}) (E + \frac{(K - Δ_{l}^{K})}{y_{3}^{2}} C)] \end{matrix}$ (38) and the notations A, B, C, E, F and G are equal to the following: $A = \frac{(h_{g} - Δ_{l}^{h_{g}}) + (h_{d} - Δ_{l}^{h_{d}})}{2 (x + Δ_{h}^{x})}$ $B = \frac{(γ + Δ_{h}^{γ}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}{(γ + Δ_{h}^{γ}) - (α - Δ_{l}^{α}) + e^{(n + Δ_{h}^{n}) (β + Δ_{h}^{β})}}$ $C = \frac{(γ - Δ_{l}^{γ}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}{(γ - Δ_{l}^{γ}) - (α + Δ_{h}^{α}) + e^{(n - Δ_{l}^{n}) (β - Δ_{l}^{β})}}$ $E = \frac{(h_{g} + Δ_{h}^{h_{g}}) + (h_{d} + Δ_{h}^{h_{d}})}{2 (x - Δ_{l}^{x})}$ $F = \frac{(h_{g} + h_{d})}{2 x}, G = \frac{γ + e^{n β}}{γ + e^{n β} - α}$

From constraints (25.6) and (25.9), it can be deducted that λ₃ = 0. If λ₁ = λ₂ = 0 in (25.7) and (25.8), then y₃ < y₂ < y₁ and this constraint is against the constraint 0 < y₁ ≤ y₂ ≤ y₃ . Therefore, y₁ = y₂ and y₂ = y₃, mean that y₁ = y₂ = y₃ = y^* . Based on this explanation, we can obtain the solution of the model by solving Equaitons (25.1) –(25.11), as follows: $\frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{1}} + \frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{2}} + \frac{\partial T \tilde{PU} ({\tilde{y}}_{n})}{\partial y_{3}} = 0$ (39)

By taking the derivations and replacing them in Equation (29), and rearranging, finally, the solution of the model is as follows: $y^{*} = \sqrt{\frac{ϒ_{1}}{ϒ_{2}}}$ (40) where $\begin{matrix} ϒ_{1} = (D - Δ_{l}^{D}) (K + Δ_{h}^{K}) B \\ + (D + Δ_{h}^{D}) (K - Δ_{l}^{K}) C + 4 KDG \\ ϒ_{2} = (D + Δ_{h}^{D}) (AC - E) + (D - Δ_{l}^{D}) (BE - A) \\ + 4 FD (G - 1) + \frac{(h_{g} - Δ_{l}^{h_{g}})}{2 C} - \frac{(h_{g} + Δ_{h}^{h_{g}})}{2 B} + 2 \frac{h_{g}}{G} \end{matrix}$

6 Numerical example

In this section, a numerical example is presented to illustrate the application of the developed model. The relevant information for this example, except x, which is considered as x = 170, 000 units per year, is taken from Wahab and Jaber [51] and consists of D = 50, 000 units/year, K = $3, 000 per order, C = $100 per unit, s = $200 per unit, v = $50 per unit, d = $0.5 per unit, h_g = $20 unit/year, h_d = $5unit/year. Moreover, the learning function parameters are the same as Wahab and Jaber [51]: γ = 819.76 α = 70.07, β = 0.79. The input parameters (D, K, s, v, d, C, x, h_d, h_g, α, γ, β) are fuzzified with arbitrary values for diferent levels of fuzziness from +10% to +70%.

Figure 1 illustrates the three-dimensional graph of the number of shipments, level of fuzziness and EOQ that are plotted for the first 30 shipments. For a constant level of fuzziness, the more the number of shipment increases the more the optimal lot size decreases. This reduction continues until it is fixed for a particular number of shipments, which is about 21 for a 30-percent level of fuzziness. Since the learning increases through the number of shipments. In addition, in Fig. 1, the optimal lot size depends directly on the level of uncertainty. It increases when the level of uncertainty of a system increases. This is also consistent with the reality, because when the uncertainty increases, the DM deals with the lower bound of knowledge about the number of defective items. Thus, s/he should order more. Therefore, it would be advantageous for both practitioners and researchers if the impreciseness of the model could be decreased to avoid costly inventory policies.

Figure 2 depicts the three-dimensional graph of the number of shipments, level of fuzziness and TPU according to the optimal lot sizes plotted in Fig. 1. This graph also gives the compatible results compared to the optimal lot sizes (Fig. 1). The uncertainty has the reverse influence on both the EOQ and TPU. That is, for a fixed level of uncertainty, the number of optimal order quantities decreases while the total profit increases. On the other hand, when the level of uncertainty increases, the order quantity and TPU increases as well as the quantity of shipments.

7 Comparison to earlier models

Yadav et al. [53, 54] presented two inventory models that are similar to the EOQ model developed in this paper. However, they considered fuzzy models where only the demand rate is fuzzified by TFN, and not only percentage of defective items follows a learning curve, but also a part of ordering and holding costs decreases because of the effect of learning. Besides, Yadav et al. [53] considered Type 1 and 2 screening errors which may occur when good items mistakenly taken as defectives ones and defective items mistakenly taken as good ones, respectively.

Figures 3 and 4 compare the optimal order quantity obtained by these models for the first 10 shipments under 10 and 70 percent level of fuzziness, respectively. Moreover, the similar patterns are presented in Figs. 5 and 6 for TPU. For a better comparison, Table 2 shows the average percentage change between the crisp and the approximated defuzzified optimal values of the first 10 shipments for different levels of fuzziness in each model separately. To calculate these results, in addition to the data in the previous section, we considered the penalty cost as $100/unit due to the screening error, and Type 1 and 2 errors 0.02 and 0.03, respectively. The constant part of holding and ordering costs are supposed to be $20/unit/year and $2700 per order while the other part of these costs which is affected by the similar learning rate 0.2 are considered as $5/unit/year and $300/order, respectively.

The results in Figs. 3 and 4 show that the optimal order quantity proposed by the fully fuzzified model of this paper is the largest of the three models for the mentioned levels of fuzziness. According to the Table 2, it is also evidenced that the increase of average percentage change of EOQ from the 10% to 70% level of fuzziness for fully fuzzified model is about 8 times more than the other ones. Similar results can be extracted for TPU. These results indicate that with increasing the level of fuzziness one of the most effective strategies to capture real inventory situations is handling the induced uncertainty by fuzzifying more parameters. The effect of learning is more tangible on the optimal order quantity for the fully fuzzified model when the level of uncertainty increases.

8 Managerial insights

Consider a buyer who sends some orders to a supplier. Each order contains a different lot and the supplier considers separate shipments to meet the buyer’s demand. The produced lots are not of perfect quality and contain defective items. They pass the screening process by some workers at the buyer. In this situation the numbers of defective items per lot may vary from one shipment to another. Therefore, the percentage of defective items per lot conforms to a degree of fuzziness. The workers screen out the initial lots without previous experience, and, consequently, the number of the optimal lot size increases. As the number of shipments increases, the knowledge of the buyer and supplier about the product quality preferences and production system intensifies. In the later shipments, the buyer could transfer the information about the quality aspects of the received product to the supplier and the supplier could modify the process or adopt some corrective actions, and, simultaneously, the learning process will occur. Therefore, although the number of defective items has an imprecise percentage, it could be reduced by a close collaboration between the buyer and the supplier.

For example, in foundry industries, the buyer may find some cavity problems in the casting parts and report this quality problem to the supplier, and, consequently, the supplier may add an X-ray operation to its production system in order to detect the cavity in the parts before sending the parts to the buyer. Adding this operation definitely decreases the number of defective items in future shipments. In this scenario, the decision-makers could order smaller batches so that the number of defective items in each batch decreases, and, consequently, they could acquire more profit. They could raise the gained profit by raising the number of the orders when uncertainty is at a high level. Through this, they could ensure that the number of items, which does not conform with the quality, declines using the learning process, while still meeting the financial target. As a managerial insight, the DM should expect an increase in the total profit when learning occurs. This strategy justifies some costs that are devoted to the learning process in the long-term.

9 Summary and conclusion

In this paper, an EOQ model with imperfect quality items under different holding costs and learning in inspection is developed in a fully fuzzy situation. The results reveal that the optimal lot size has a descending order and declines continuously as the shipment increases while optimal total profit increases by growth in the number of shipments.

There are considerable extensions for future studies. One immediate extension is to apply different types of learning curve to evaluate the behavior of the model as well as potential alternation in inventory policies. Future research could also develop the model by a combination of the learning and fuzziness in other parameters of the models, such as demand, screening rate, holding cost of good and defective items. In such a scenario, the study of the impact of learning curves with different rates of learning on fuzzified parameters, and, finally, on the total profit function, seems to be another interesting area. Also, future researches could employ learning to decrease the degree of fuzziness, as proposed by Kazemi et al. [32, 33].

Footnotes

Acknowledgments

The first author wishes to express his gratitude to University of Malaya for funding his research (Grant No. RP018b-13aet). The authors are also immensely grateful to the Associate Editor, Prof. Jian Wu, and anonymous reviewers for the comments and suggestions that helped to improve the paper.

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